Question: Determine how many solutions exist for the system of equations. ${4x+y = 10}$ ${-x+y = 9}$
Answer: Convert both equations to slope-intercept form: ${4x+y = 10}$ $4x{-4x} + y = 10{-4x}$ $y = 10-4x$ ${y = -4x+10}$ ${-x+y = 9}$ $-x{+x} + y = 9{+x}$ $y = 9+x$ ${y = x+9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+10}$ ${y = x+9}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.